Now let's move on to g(x) = sin(2x) ex2. We
could, of course, take derivatives at x = 0, but because of
product and chain rule they quickly get out of hand. Therefore, we will
try another method: multiplication with an "infinite" distributive law.
We know:

sin(x) =
ex =

Substituting and multiplying we therefore get:

sin(2x) ex2 =
=
=
=
=
=
=
=
=

This approximation is not bad as you can see when you overlay the two
functions. The red curve below shows the original function, the blue one shows
the first three terms from the Taylor serires.

Of course the polynomial is much simpler than the original
function, yet both functions are pretty close to each other if
x is small.